Free Bonferroni Correction Calculator

When multiple statistical tests are performed on the same dataset, the chance of at least one false‑positive result (Type I error) grows quickly. The Bonferroni correction (often called the Bonferroni adjustment) is a widely used multiple comparisons correction technique that keeps the overall error rate in check by lowering the significance threshold for each individual test. This article explains the rationale behind the method, shows how to derive the adjusted significance level using both the classic formula and the Šidák correction, and discusses the practical trade‑offs of using this approach.

What Is the Bonferroni Correction?

The Bonferroni correction addresses the problem of multiplicity by dividing the desired family‑wise significance level $\alpha$ by the number of comparisons $m$. If $\alpha = 0.05$ and ten hypotheses are tested, each test is evaluated against the adjusted threshold $\alpha_{\text{adj}} = 0.05 / 10 = 0.005$. A result is declared statistically significant only if its p‑value is less than this corrected value. This simple Bonferroni adjustment keeps the probability of any single false positive at or below the original $\alpha$ level.

$$\alpha_{\text{adj}} = \dfrac{\alpha}{m}$$

The same principle applies whether you are performing post‑hoc comparisons in ANOVA, testing multiple endpoints in clinical trials, or running many correlations on one sample.

How to Calculate the Corrected Threshold

You can choose between the classic Bonferroni formula and the Šidák correction, which is slightly less conservative when the tests are independent.

Classic Bonferroni

Divide the original significance level $\alpha$ by the number of tests $m$.

$$\alpha_{\text{adj}} = \dfrac{\alpha}{m}$$

Šidák Correction

Assuming independence, the Šidák method computes the adjusted level as:

$$\alpha_{\text{adj}} = 1 - (1 - \alpha)^{\frac{1}{m}}$$

Worked Example

• Number of tests ($m$): 4
• Nominal significance level ($\alpha$): 0.05 (5%)

Classic Bonferroni:

$$\alpha_{\text{adj}} = \dfrac{0.05}{4} = 0.0125$$

Each test must achieve a p‑value below 0.0125 to be considered significant.

Šidák correction:

$$\alpha_{\text{adj}} = 1 - (1 - 0.05)^{\frac{1}{4}} = 1 - 0.95^{0.25} \approx 0.0127$$

In this case the threshold is very close to the classic result (0.0127 vs. 0.0125), but the difference becomes larger as $m$ increases or $\alpha$ is lowered.

Why the Correction Is Important

Without any adjustment, the family‑wise error rate (FWER) grows as $1 - (1 - \alpha)^m$. For 20 tests at $\alpha = 0.05$, there is a $1 - 0.95^{20} \approx 64\%$ chance of encountering at least one false positive. The Bonferroni correction forces the FWER to remain at or below $\alpha$, making the significant findings more trustworthy when multiple comparisons are carried out.

Limitations and Alternatives

Despite its simplicity, the Bonferroni correction can be overly conservative, especially when many tests are run or the tests are highly correlated. This conservatism increases the probability of false negatives (Type II errors), meaning that genuinely important effects may be missed.

For this reason, researchers often turn to less strict procedures:

• Holm‑Bonferroni method – sequentially adjusts p‑values while maintaining strong control of the FWER.
• Benjamini‑Hochberg procedure – controls the false discovery rate (FDR) rather than the FWER, offering higher power when many true effects are expected.

The choice between these approaches depends on the research context and how the investigator balances false positives against false negatives. The Bonferroni correction remains the most straightforward and universally understood option, making it a solid starting point for any multiple comparisons correction routine.

This tool automates the computation of both the classic Bonferroni adjustment and the Šidák correction, letting you quickly obtain the adjusted significance level for any number of comparisons and any nominal $\alpha$.